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Energetics of protein structures
• Molecular Mechanics force fields
• Implicit solvent
• Statistical potentials
Energetics of protein structures
• Molecular Mechanics force fields
• Implicit solvent
• Statistical potentials
What is an atom?
• Classical mechanics: a solid object
• Defined by its position (x,y,z), its shape (usually a ball) and its mass
• May carry an electric charge (positive or negative), usually partial (less than an electron)
MASS 20 C 12.01100 C ! carbonyl C, peptide backboneMASS 21 CA 12.01100 C ! aromatic CMASS 22 CT1 12.01100 C ! aliphatic sp3 C for CHMASS 23 CT2 12.01100 C ! aliphatic sp3 C for CH2MASS 24 CT3 12.01100 C ! aliphatic sp3 C for CH3MASS 25 CPH1 12.01100 C ! his CG and CD2 carbonsMASS 26 CPH2 12.01100 C ! his CE1 carbonMASS 27 CPT 12.01100 C ! trp C between ringsMASS 28 CY 12.01100 C ! TRP C in pyrrole ring
Example of atom definitions: CHARMM
RESI ALA 0.00GROUP ATOM N NH1 -0.47 ! |ATOM HN H 0.31 ! HN-NATOM CA CT1 0.07 ! | HB1ATOM HA HB 0.09 ! | /GROUP ! HA-CA--CB-HB2ATOM CB CT3 -0.27 ! | \ATOM HB1 HA 0.09 ! | HB3ATOM HB2 HA 0.09 ! O=CATOM HB3 HA 0.09 ! |GROUP !ATOM C C 0.51ATOM O O -0.51BOND CB CA N HN N CA BOND C CA C +N CA HA CB HB1 CB HB2 CB HB3 DOUBLE O C
Example of residue definition: CHARMM
Forces between atoms
Strong bonded interactions
20 )( bbKU
20)( KU
))cos(1( nKU
b
All chemical bonds
Angle between chemical bonds
Preferred conformations forTorsion angles: - angle of the main chain - angles of the sidechains
(aromatic, …)
Forces between atoms: vdW interactions
612
2)(r
R
r
RrE ijij
ijLJ
1/r12
1/r6
Rij
r
Lennard-Jones potential
jiijji
ij
RRR
;
2
Some Common force fields in Computational Biology
ENCAD (Michael Levitt, Stanford)
AMBER (Peter Kollman, UCSF; David Case, Scripps)
CHARMM (Martin Karplus, Harvard)
OPLS (Bill Jorgensen, Yale)
MM2/MM3/MM4 (Norman Allinger, U. Georgia)
ECEPP (Harold Scheraga, Cornell)
GROMOS (Van Gunsteren, ETH, Zurich)
Michael Levitt. The birth of computational structural biology. Nature Structural Biology, 8, 392-393 (2001)
Energetics of protein structures
• Molecular Mechanics force fields
• Implicit solvent
• Statistical potentials
Potential of mean force
dXdYe
eYXP
kT
YXU
kT
YXU
,
,
),(
A protein in solution occupies a conformation X with probability:
X: coordinates of the atoms of the protein
Y: coordinates of the atoms of the solvent
),()()(),( YXUYUXUYXU PSSP
The potential energy U can be decomposed as: UP(X): protein-protein interactions
US(X): solvent-solvent interactions
UPS(X,Y): protein-solvent interactions
Potential of mean force
dYYXPXPP ),()(
We study the protein’s behavior, not the solvent:
PP(X) is expressed as a function of X only through the definition:
dXe
eXP
kT
XW
kT
XW
P T
T
)(
)(
)(
WT(X) is called the potential of mean force.
Potential of mean force
The potential of mean force can be re-written as:
)()()( XWXUXW solPT
Wsol(X) accounts implicitly and exactly for the effect of the solvent on the protein.
Implicit solvent models are designed to provide an accurate and fast
estimate of W(X).
The SA model
Surface area potential
N
kkkvdWcav SAWW
1
Eisenberg and McLachlan, (1986) Nature, 319, 199-203
Hydrophobic potential:Surface Area, or Volume?
(Adapted from Lum, Chandler, Weeks, J. Phys. Chem. B, 1999, 103, 4570.)
“Radius of the molecule”
Volume effect
Surface effect
For proteins and other large bio-molecules, use surface
Protein Electrostatics
• Elementary electrostatics• Electrostatics in vacuo• Uniform dielectric medium• Systems with boundaries
• The Poisson Boltzmann equation• Numerical solutions• Electrostatic free energies
• The Generalized Born model
Elementary Electrostatics in vacuo
Some basic notations:
2
2
2
2
2
22
z
f
y
f
x
fffgraddivf
z
f
y
f
x
ffgradf
z
F
y
F
x
FFdivF zyx
Divergence
Gradient
Laplacian
Elementary Electrostatics in vacuo
ur
qqF
20
21
4
Coulomb’s law:
The electric force acting on a point charge q2 as the result of the presence ofanother charge q1 is given by Coulomb’s law:
q1
r
u
Electric field due to a charge:
By definition:
ur
q
q
FE
20
1
2 4
q2
q1
E “radiates”
Elementary Electrostatics in vacuo
0
qdAE
0
)())((
X
X Ediv
Gauss’s law:
The electric flux out of any closed surface is proportional to the total charge enclosed within the surface.
Integral form: Differential form:
Notes:- for a point charge q at position X0, (X)=q(X-X0)
- Coulomb’s law for a charge can be retrieved from Gauss’s law
Elementary Electrostatics in vacuo
UgradF
gradE
Energy and potential:
- The force derives from a potential energy U:
- By analogy, the electric field derives from an electrostatic potential :
For two point charges in vacuo:
r
qqU
0
21
4
r
q
0
1
4
Potential produced by q1 atat a distance r:
Elementary Electrostatics in vacuo
The cases of multiple charges: the superposition principle:
Potentials, fields and energy are additive
For n charges:
ji i
ji
N
ii
i
i
N
i i
i
XX
qqU
uXX
qXE
XX
qX
0
12
0
1 0
4
4)(
4
q1
q2
qi
qN
X
Elementary Electrostatics in vacuo
0
2
0
graddiv
Ediv
Poisson equation:
Laplace equation:
02 (charge density = 0)
+-
Uniform Dielectric MediumPhysical basis of dielectric screening
An atom or molecule in an externally imposed electric field develops a nonzero net dipole moment:
(The magnitude of a dipole is a measure of charge separation)
The field generated by these induced dipoles runs against the inducingfield the overall field is weakened (Screening effect)
The negativecharge is screened bya shell of positivecharges.
+
Uniform Dielectric MediumElectronic polarization:
- ---
----
----
-
+
- ----
----
---
-Under external
field
Resulting dipole moment
Orientation polarization:
Under externalfield
Resulting dipole moment
Uniform Dielectric MediumPolarization:
The dipole moment per unit volume is a vector field known asthe polarization vector P(X).
In many materials: )(4
1)()( XEXEXP
is the electric susceptibility, and is the electric permittivity, or dielectric constant
The field from a uniform dipole density is -4P, therefore the total field is
applied
applied
EE
PEE
4
Uniform Dielectric Medium
Some typical dielectric constants:
Molecule Dipole moment (Debyes) of a
single molecule
Dielectric constant of the
liquid at 20°C
Water 1.9 80
Ethanol 1.7 24
Acetic acid 1.7 4
Chloroform 0.86 4.8
Uniform Dielectric Medium
Modified Poisson equation:
0
2 graddiv
Energies are scaled by the same factor. For two charges:
r
qqU
0
21
4
Uniform Dielectric Medium
The work of polarization:
It takes work to shift electrons or orient dipoles. A single particle with charge q polarizes the dielectric medium; there is areaction potential that is proportional to q for a linear response.
The work needed to charge the particle from qi=0 to qi=q:
q
Rii
q
iiR qCqdqqCdqqW0
2
0 2
1
2
1
CqR
For N charges:
N
iiRiqW
12
1 Free energy
System with dielectric boundaries
The dielectric is no more uniform: varies, the Poisson equation becomes:
0
)()(
XXXXgradXdiv
If we can solve this equation, we have the potential, from which we can derivemost electrostatics properties of the system (Electric field, energy, free energy…)
BUT
This equation is difficult to solve for a system like a macromolecule!!
The Poisson Boltzmann Equation
(X) is the density of charges. For a biological system, it includes the chargesof the “solute” (biomolecules), and the charges of free ions in the solvent:
The ions distribute themselves in the solvent according to the electrostatic potential (Debye-Huckel theory):
N
iiiions XnqX
1
)()(
ion i typeon charge :
eunit volumper i typeof ions ofnumber :
i
i
q
nkT
Xq
i
ii
en
Xn )(
0
)(
)()()( XXX ionssolute
The potential f is itself influenced by the redistribution of ion charges, so thepotential and concentrations must be solved for self consistency!
N
i
kT
Xq
ii
i
enqX
XX1
)(0
00
1)(
The Poisson Boltzmann Equation
Linearized form:
IkT
qnkT
XXXX
XX
N
iii
01
20
0
2
2
0
21
)()()()(
I: ionic strength
• Analytical solution
• Only available for a few special simplification of the molecular shape and charge distribution
• Numerical Solution• Mesh generation -- Decompose the physical
domain to small elements;• Approximate the solution with the potential value
at the sampled mesh vertices -- Solve a linear system formed by numerical methods like finite difference and finite element method
• Mesh size and quality determine the speed and accuracy of the approximation
Solving the Poisson Boltzmann Equation
Linear Poisson Boltzmann equation:Numerical solution
P
w
• Space discretized into a cubic lattice.
• Charges and potentials are defined on grid points.
• Dielectric defined on grid lines
• Condition at each grid point:
6
1
22
0
6
1
jijijij
i
jjij
i
h
hq
j : indices of the six direct neighbors of i
Solve as a large system of linearequations
Electrostatic solvation energy
The electrostatic solvation energy can be computed as an energy change when solvent is added to the system:
i
NSSii
RFielec iiqiqW )()(
2
1)(
2
1
The sum is over all nodes of the lattice
S and NS imply potentials computed in the presence and absence of solvent.
Approximate electrostatic solvation energy:The Generalized Born Model
Remember:
N
i
RFiielec qG
12
1
For a single ion of charge q and radius R:
1
1
8 0
2
R
qGBorn Born energy
For a “molecule” containing N charges, q1,…qN, embedded into spheres or radii R1, …, RN such that the separation between the charges is large compared to the radii, the solvation energy can be approximated by the sum of the Born energy and Coulomb energy:
N
i
N
ij ij
jiN
i i
ielec r
R
qG
1 01 0
2
11
42
11
1
8
Approximate electrostatic solvation energy:The Generalized Born Model
The GB theory is an effort to find an equation similar to the equation above, that is a good approximation to the solution to the Poisson equation.The most common model is:
N
i
N
jaa
r
jiij
jiGB
ji
ij
eaar
qqG
1 142
02
11
8
1
ai: Born radius of charge i:
GGB is correct when rij 0 and rij ∞
Assuming that the charge i produces a Coulomb potential:
iRr
i r
dV
a 44
11
Further reading
• Michael Gilson. Introduction to continuum electrostatics. http://gilsonlab.umbi.umd.edu
• M Schaefer, H van Vlijmen, M Karplus (1998) Adv. Prot. Chem., 51:1-57 (electrostatics free energy)
• B. Roux, T. Simonson (1999) Biophys. Chem., 1-278 (implicit solvent models)
• D. Bashford, D Case (2000) Ann. Rev. Phys. Chem., 51:129-152 (Generalized Born models)
• K. Sharp, B. Honig (1990) Ann. Rev. Biophys. Biophys. Chem., 19:301-352 (Electrostatics in molecule; PBE)
• N. Baker (2004) Methods in Enzymology 383:94-118 (PBE)
Energetics of protein structures
• Molecular Mechanics force fields
• Implicit solvent
• Statistical potentials